On incidence coloring and star arboricity of graphs.

*(English)*Zbl 0871.05022Summary: We show that the concept of incidence coloring introduced by R. A. Brualdi and J. J. Q. Massey [(*) Discrete Math. 122, No. 1-3, 51-58 (1993; Zbl 0790.05026)] is a special case of directed star arboricity, introduced by I. Algor and N. Alon [(**) Discrete Math. 75, No. 1-3, 11-22 (1989; Zbl 0684.05033)]. A conjecture in (*) concerning asymptotics of the incidence coloring number is solved in the negative following an example in (**). We generalize a result in [N. Alon, C. McDiarmid, and B. Reed, Combinatorica 12, No. 4, 375-380 (1992; Zbl 0780.05043)] concerning the star arboricity of graphs to the directed case by a slight modification of their proof, to give the same asymptotic bound as in the undirected case. As a result, we get tight asymptotic bounds for the maximum incidence coloring number of a graph in terms of its degree.

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\textit{B. Guiduli}, Discrete Math. 163, No. 1--3, 275--278 (1997; Zbl 0871.05022)

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##### References:

[1] | Akiyama, J; Kano, M, Path factors of a graph, () · Zbl 0605.05034 |

[2] | Algor, I; Alon, N, The star arboricity of graphs, Discrete math., 75, 11-22, (1989) · Zbl 0684.05033 |

[3] | Alon, N; McDiarmid, C; Reed, B, Star arboricity, Combinatorica, 12, 375-380, (1992) · Zbl 0780.05043 |

[4] | Brualdi, R.A; Massey, J.J.Q, Incidence and strong edge colorings of graphs, Discrete math., 122, 51-58, (1993) · Zbl 0790.05026 |

[5] | Erdős, P; Lovász, L, Problems and results on 3-chromatic hypergraphs and some related questions, () · Zbl 0315.05117 |

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